In this article, we prove that any Q-factorial weak Fano 3-fold with only terminal singularities has a smoothing. 0. Introduction Definition 0.1. Let X be a normal Gorenstein projective variety of dimension 3 over C which has only terminal singularities. (1) If −KX is ample, we call X a Fano 3-fold. (2) If −KX is nef and big, we call X a weak Fano 3-fold. Definition 0.2. Let X be a normal Goren...

Let $(R,fm,k)$ be a local Gorenstein ring of dimension $n$. Let $H_{I,J}^i(R)$ be the  local cohomology with respect to a pair of ideals $I,J$ and $c$ be the $inf{i|H_{I,J}^i(R)neq0}$. A pair of ideals $I, J$ is called cohomologically complete intersection if $H_{I,J}^i(R)=0$ for all $ineq c$. It is shown that, when $H_{I,J}^i(R)=0$ for all $ineq c$, (i) a minimal injective resolution of $H_{I,...

In this paper, we introduce a new homological invariant called quasi-projective dimension, which is generalization of projective dimension. We discuss various properties Among other things, prove the following. (1) Over quotient regular local ring by sequence, every finitely generated module has finite (2) The Auslander--Buchsbaum formula and depth for modules dimension remain valid (3) Several...

By a symplectic manifold (or a symplectic n-fold) we mean a compact Kaehler manifold of even dimension n with a non-degenerate holomorphic 2form ω, i.e. ω is a nowhere-vanishing n-form. This notion is generalized to a variety with singularities. We call X a projective symplectic variety if X is a normal projective variety with rational Gorenstein singularities and if the regular locus U of X ad...

A well-known conjecture says that every one-relator group is coherent. We state and partly prove a similar statement for graded associative algebras. In particular, we show that every Gorenstein algebra A of global dimension 2 is graded coherent. This allows us to define a noncommutative analogue of the projective line P as a noncommutative scheme based on the coherent noncommutative spectrum q...